127 (2002)
MATHEMATICA BOHEMICA
No. 2, 181–195
Proceedings of EQUADIFF 10, Prague, August 27–31, 2001
QUALITATIVE THEORY OF HALF-LINEAR SECOND ORDER
DIFFERENTIAL EQUATIONS
Ondřej Došlý, Brno
Dedicated to the memory of Prof. Árpád Elbert
Abstract. Some recent results concerning properties of solutions of the half-linear second
order differential equation
(∗)
(r(t)Φ(x )) + c(t)Φ(x) = 0,
Φ(x) := |x|p−2 x,
p > 1,
are presented. A particular attention is paid to the oscillation theory of (∗). Related
problems are also discussed.
Keywords: half-linear equation, Picone’s identity, scalar p-Laplacian, variational method,
Riccati technique, principal solution
MSC 2000 : 34C10
1. Introduction
In this contribution we deal with the oscillatory properties and related problems
concerning the half-linear second order differential equation
(1)
(r(t)Φ(x )) + c(t)Φ(x) = 0,
Φ(x) := |x|p−2 x,
p > 1,
where r, c are continuous functions and r(t) > 0. During the recent years it was
shown that solutions of (1) behave in many aspects like those of the Sturm-Liouville
equation
(2)
(r(t)x ) + c(t)x = 0
Research supported by Grant A1019902 of the Grant Agency of the Czech Academy of
Sciences.
181
which is the special case p = 2 of (1). The aim of this paper is to present some results
of this investigation and also to point out situations where the properties of (1) and
(2) (considerably) differ. Note that the term half-linear equations is motivated by
the fact that the solution space of (1) has just one half of the properties which
characterize linearity, namely homogeneity (but not additivity).
The investigation of qualitative properties of nonlinear second order differential
equations has a long history. Recall here only the papers of Emden [26], Fowler [27],
Thomas [40], and the book of Sansone [39] containing the survey of the results
achieved in the first half of the last century. In the fifties and later decades the
number of papers devoted to nonlinear second order differential equations increased
rapidly, so we mention here only treatments directly associated with (1). Even if
some ideas concerning the properties of solutions of (1) can be already found in
the papers of Bihari [4], [5], Elbert and Mirzov with their papers [21], [36] are the
ones usually regarded as pioneers of the qualitative theory of (1). In later years, in
particular in the nineties, the striking similarity between oscillatory properties of (1)
and (2) was revealed. On the other hand, in some aspects, e.g. the Fredholm-type
alternative for solutions of boundary value problems associated with (1), it turned
out that the situation is completely different in the linear and half-linear case and the
absence of additivity of the solution space of (1) brings completely new phenomena.
The paper is organized as follows. In the next section we present a brief survey of
basic properties of solutions of (1). Section 3 is devoted to the oscillation theory of
(1) and in the last section we discuss some other problems associated with (1).
2. Basic properties of (1)
Consider a special equation of the form (1)
(Φ(x )) + (p − 1)Φ(x) = 0
and denote by S = S(t) its solution given by the initial condition S(0) = 0, S (0) = 1.
In [21] it is shown that the behaviour of this solution is very similar to that of the
classical sine function. In particular, this function is odd, periodic with the half2
p
period πp := p sin(
/p) , and satisfies the generalized Pythagorian identity |S(t)| +
p
|S (t)| = 1.
Using this function, one can introduce the generalized Prüfer transformation as
follows. Let x be a nontrivial solution of (1). There exist differentiable functions , ϕ
such that
x(t) = (t)S(ϕ(t)),
182
rq−1 (t)x (t) = (t)S (ϕ(t)),
where q is the conjugate number of p, i.e., p1 + 1q = 1, and the functions , ϕ satisfy
a certain first order system with a Lipschitzian right-hand side, which is, in turn,
uniquely solvable. Hence, the same holds for (1): given t0 ∈ Ê and A, B ∈ Ê, there
exists a unique solution of (1) satisfying x(t0 ) = A, x (t0 ) = B which is extensible
over the whole interval where r, c are continuous and r(t) > 0.
Other important objects associated with (1) are the p-degree functional
(3)
b
F (y; a, b) :=
[r(t)|y |p − c(t)|y|p ] dt
a
(equation (1) is the Euler-Lagrange equation of F considered in the class of functions
satisfying the zero boundary condition y(a) = 0 = y(b)) and the generalized Riccati
equation
(4)
w + c(t) + (p − 1)r1−q (t)|w|q = 0,
q=
p
,
p−1
which is related to (1) by the substitution w = rΦ(x )/Φ(x).
Functional (3) and equation (4) are related by the half-linear version of Picone’s
identity
b
F (y; a, b) = w(t)|y||a + p
where
P (u, v) :=
b
r1−q (t)P (rq−1 (t)y , w(t)Φ(y)) dt,
a
|u|p
|v|q
− uv +
0
p
q
for all u, v ∈ Ê with the equality if and only if v = Φ(u), and w is a solution of (4)
defined on the whole interval [a, b]. This identity is the main tool in the proof of
the next statement which summarizes the basic oscillatory properties of (1). This
statement is usually referred as the Roundabout Theorem.
Proposition 1. The following statements are equivalent:
(i) Equation (1) is disconjugate on an interval I = [a, b], i.e., any nontrivial solution
of (1) has at most one zero in I.
(ii) There exists a solution of (1) having no zero in [a, b].
(iii) There exists a solution w of the generalized Riccati equation (4) which is defined
on the whole interval [a, b].
(iv) The p-degree functional F (y; a, b) is positive for every 0 ≡ y ∈ W01,p (a, b).
Observe that Proposition 1 implies that Sturm separation and comparison theorems extend verbatim to (1). Indeed, the separation theorem is essentially the
183
equivalence (i) ⇐⇒ (ii), while the comparison theorem is hidden in the equivalence
(i) ⇐⇒ (iv). In particular, similarly as in the linear case, equation (1) can be classified as oscillatory or nonoscillatory according to whether any nontrivial solution
has or has not infinitely many zeros tending to ∞.
Finally, let us mention at least two differences in the basic properties of solutions of
(1) (in addition to the already mentioned absence of additivity of the solution space).
If x1 , x2 is a pair of linearly independent solutions of (1), we have no analogue of the
Wronskian-type identity r(x1 x2 − x1 x2 ) = const which holds for (2), see [22]. We
have also no half-linear analogue of the linear transformation identity with x = h(t)y,
(5)
h(t)[(r(t)x ) + c(t)x] = (r(t)h2 (t)y ) + h(t)[(r(t)h (t)) + c(y)h(t)]y,
which is the starting point of the transformation theory of (2), see [6].
3. Oscillation theory
The equivalences given in the Roundabout Theorem (Proposition 1) suggest two
main methods of the oscillation theory of (1). The basic ideas and results based on
them are briefly explained in this section.
3.1.
. This method is based on the equivalence (i) ⇐⇒
(iv) in Proposition 1. According to this equivalence, to prove that (1) is oscillatory
it suffices to construct (for any T ∈ Ê) a nontrivial function y ∈ W01,p (T, ∞) such
that
∞
[r(t)|y |p − c(t)|y|p ] dt 0.
(6)
F (y; T, ∞) :=
T
On the other hand, for nonoscillation of (1) we need to show the existence of T ∈ Ê
such that F (y; T, ∞) > 0 for every 0 ≡ y ∈ W01,p (T, ∞).
In oscillation criteria, a typical construction of a function y for which (6) holds
reads as follows. Let T be arbitrary, T < t0 < t1 < t2 < t3 , and let

t ∈ [T, t0 ],

 0,




f (t), t ∈ [t0 , t1 ],


(7)
y(t) = 1,
t ∈ [t1 , t2 ],



 g(t), t ∈ [t2 , t3 ],




0,
t ∈ [t3 , ∞),
where f, g are solutions of the one-term equation
(8)
184
(r(t)Φ(x )) = 0
satisfying f (t0 ) = 0, f (t1 ) = 1, g(t2 ) = 1, g(t3 ) = 0, i.e.
t
f (t) =
g(t) =
r1−q (s) ds
t0
t3
t1
r1−q (s) ds
−1
t0
r
1−q
t3
r
(s) ds
t
1−q
,
−1
(s) ds
.
t2
By direct computation, using the second mean value theorem of integral calculus
t
t
applied to the integrals t01 r|f |p , t23 r|g|p (see e.g. [14]) we have
(9)
t1
F (y; T, ∞) =
r
1−q
1−p
(s) ds
t3
r
+
t0
1−q
1−p
(s) ds
s2
+
t2
c(t) dt,
s1
where s1 ∈ (t0 , t1 ), s2 ∈ (t2 , t3 ). Using this computation we can now easily prove the
following oscillation criteria.
∞ 1−q
Theorem 1. Suppose that
r
(t) dt = ∞. Then each of the following conditions is sufficient for oscillation of (1):
(i) (Leighton-Wintner-type criterion [36]).
(10)
b
lim
b→∞
c(t) = ∞.
(ii) (Nehari-type criterion [14]). The integral
(11)
lim inf
t→∞
t
r1−q (s) ds
∞
p−1 c(t) dt is convergent and
∞
c(s) ds
> 1.
t
.
A short computation and (9) show that each of conditions (i), (ii) implies that the points ti , i = 0, . . . , 3, can be chosen in such a way that F (y; T, ∞) < 0.
Concerning the nonoscillation criteria proved via the variational method, their
proofs are usually based on the Wirtinger-type inequality
∞
(12)
T
|M (t)||y|p dt pp
∞
T
M p (t)
|y |p dt
|M (t)|p−1
where M is a differentiable function such that M (t) = 0 on [T, ∞), which holds
for every 0 ≡ y ∈ W01,p (T, ∞), see e.g. [12]. Using (12) we can prove the following
statement.
185
Theorem 2. Suppose that
(13)
lim sup
t
∞
r1−q (s) ds
t→∞
r1−q (t) dt = ∞ and
p−1 ∞
c+ (s) ds
<
t
p−1
1 p−1
,
p
p
where c+ (t) = max{0, c(t)}. Then (1) is nonoscillatory.
t
. Setting M (t) := ( r1−q (s) ds)1−p and using (13) one can show that
F (y; T, ∞) < 0.
The previous two statements are typical examples of the application of the variational method in the oscillation theory of half-linear equations.
3.2.
. In this subsection we briefly sketch how the equivalence (i) ⇐⇒ (iii) can be used to derive (non)oscillation criteria. We illustrate the
application of this method in the criteria which are similar to those presented in the
previous subsection.
We start with an improvement of the statement given in Theorem 1 (ii).
∞ 1−q
∞
Theorem 3. Suppose that
r (t) dt = ∞, the integral
c(t) dt is converp−1
gent and lim inf in (11) is > 1p ( p−1
)
.
Then
(1)
is
oscillatory.
p
.
By contradiction, suppose that (1) is nonoscillatory. Then the solutions
of the associated Riccati equation (4) satisfy the integral equation
w(t) =
t
∞
c(t) dt + (p − 1)
∞
r1−q (s)|w|q ds,
t
t 1−q
p−1
see [34]. Multiplying this equation by
r
(s) ds
, using (13) and supposing
t 1−q
p−1
that lim sup
r (s) ds
w(t) =: µ < ∞ (in the case µ = ∞, to get a contrat→∞
diction is even easier than in our case µ < ∞) we find an ε > 0 such that µ satisfies
the inequality
p−1
1 p−1
µ>
+ ε + |µ|q .
p
p
p−1
0 for t ∈ Ê, we have the required contradiction.
Since |t|q − t + p1 p−1
p
Nonoscillation criteria based on the Riccati technique are usually proved using a
slightly different method than just the equivalence mentioned in Proposition 1. We
look for a solution of the inequality
(14)
186
v + c(t) + (p − 1)r1−q (t)|v|q 0
instead of (4), since it essentially means that a certain Sturmian majorant of (1) is
nonoscillatory and hence (1) is nonoscillatory as well.
Theorem 4. Suppose that
(15)
(16)
∞
r1−q (t) dt = ∞ and
p−1 ∞
c(t) dt converges. If
p−1
1 p−1
lim sup
r
(s) ds
c(s) ds <
,
p
p
t→∞
t
p−1 ∞
t
p−1
2p − 1 p − 1
lim inf
r1−q (s) ds
c(s) ds > −
,
t→∞
p
p
t
t
1−q
∞
then (1) is nonoscillatory.
.
t 1−q
1−p
p−1
Let v(t) = β
r (s) ds
, β := p1 ( p−1
. Using (15) and (16)
p )
it is not difficult to verify that this function really satisfies (14), see [14].
∞ 1−q
All criteria presented by Theorems 1–4 contain the assumption
r
= ∞. A
slight modification of the proofs of these statements enables us to find their counter ∞ 1−q
parts in the case
r
< ∞, see [14].
3.3.
. In the previous criteria, equation (1) has been regarded as a perturbation of the (nonoscillatory) one-term differential equation (8). It was shown that if the “perturbation”
function c in (1) is “sufficiently positive” (“not too positive”) then (1) becomes oscillatory (remains nonoscillatory). The exact quantitative characterization of these
vague concepts is just the content of Theorems 1–4.
From this point of view, it is a natural idea to consider (1) not as a perturbation
of the one-term equation (8), but as a perturbation of the the general two-term
nonoscillatory equation
(17)
(r(t)Φ(x )) + c̃(t)Φ(x) = 0,
where c̃ is a continuous function, and (non)oscillation criteria for (1) are formulated
in terms of the difference c− c̃. Note that this idea applied to linear equation (2) does
not produce essentially new criteria, due to the transformation identity (5). Indeed,
this identity applied to (2) written in the form (rx ) + cx = (rx ) + c̃x + (c − c̃)x = 0
and with the transformation function h which a solution of (rx ) + c̃x = 0 gives
h(t) (r(t)x ) + c̃(t)x + (c(t) − c̃(t))x = (r(t)h2 (t)y ) + h2 (t)[c(t) − c̃(t)]y,
so the resulting equation is again an equation of the form (2). One can then investigate it as the perturbation of the one-term equation (r(t)h2 (t)y ) = 0 and to
transform the results obtained “back” to the original equation (2).
187
Concerning the half-linear equation (1), as mentioned in Section 2, we have no halflinear version of (5), so the idea to investigate (1) as a perturbation of two-terms
equation (17) brings really qualitatively new criteria. This approach was (implicitly)
used for the first time by Elbert [23], who proved that (1) with r ≡ 1 is oscillatory
provided
(18)
lim
b→∞
b
γ
c(t) − p tp−1 dt = ∞,
t
γ=
p−1
p
p
.
In this setting equation (1) with r ≡ 1 is viewed as a perturbation of the generalized
Euler equation with the critical constant
(Φ(x )) +
(19)
γ
Φ(x) = 0,
tp
p−1
whose one solution x(t) = t p can be computed explicitly and the other solutions
p−1
2
behave asymptotically as t p lg p t, see [24].
A natural question is why it is just the power tp−1 which appears by the difference
c(t) − γ̃t−p in (18). In the linear case p = 2 the answer is that h = t1/2 is the so
called principal solution of the Euler equation x + 4t12 x = 0 (principal solutions
of (2) and (1) are discussed in more detail later in this section). More precisely,
the transformation x = t1/2 y transforms (2) with r ≡ 1 into the equation (ty ) +
c − 4t12 ty = 0.
The concept of the principal solution of (1) was introduced by Mirzov [37] and in
[13], [25] it was shown that this solution has many of the properties of the principal
solution of (2). Using this concept we can now prove the following generalization of
the Elbert criterion (18), see [14], [17].
Theorem 5. Suppose that (17) is nonoscillatory and h is its (positive) principal
solution. If
(20)
lim
b→∞
b
(c(t) − c̃(t)) hp (t) dt = ∞
then (1) is oscillatory.
. The proof is similar to that of Theorem 1. We modify the test function
(7) as follows. We let y = h for t ∈ [t1 , t2 ]. Then f, g are solutions of (17) (instead
of (8)) satisfying f (t1 ) = h(t1 ), g(t2 ) = h(t2 ). Using (20) and the properties of
principal solutions of (1) one can show that the points ti , i = 0, . . . , 3, can be chosen
in such a way that F (y; T, ∞) < 0, see [17].
188
Observe that in the case c̃ ≡ 0 Theorem 5 reduces to Theorem 1 (i). Indeed, if
∞ 1−q
r
= ∞ then h ≡ 1 is the principal solution of the one term equation (8) and
(20) is the same as (10). Note also that Theorems 2–4 can be extended along the
line treated in this subsection as well, we refer to [12], [17] for more details.
We conclude this part with some remarks and open problems concerning the principal solution of (1). In the linear case the principal solution x̃ of (2) is (equivalently)
defined as a solution for which
(21)
x̃(t)
=0
t→∞ x(t)
lim
⇐⇒
∞
dt
= ∞,
r(t)x̃2 (t)
where x is any solution of (2) linearly independent of x̃. Since both these characterizations are based on the linearity of the solutions space of (2), they do not extend
(directly) to (1). Mirzov [37] in his construction of the principal solution of (1) used
the fact that if this equation is nonoscillatory, then among all solutions of the associated Riccati equation (4) there exists a minimal one w,
in the sense that any other
solution w of this equation satisfies w(t) > w(t)
eventually. The principal solution
of (1) is then defined by
x̃(t) = C exp
t
r1−q (s)Φ−1 (w(s))
ds ,
Φ−1 being the inverse function of Φ, i.e. the solution x̃ which is determined by w
=
rΦ(x̃ )/Φ(x̃). In the linear case this construction is equivalent to (21), see [28].
In [16] we have tried to find an integral characterization of the principal solution
of (1) in such a way that in the linear case it reduces to the second expression in
(21). The main results of [16] are summarized in the next statement.
Theorem 6. Suppose that equation (1) is nonoscillatory and x̃ is its solution
such that x̃ (t) = 0 for large t.
(i) Let p ∈ (1, 2). If
(22)
∞
dt
r(t)x2 (t)|x (t)|p−2
= ∞,
then x̃ is the principal solution.
(ii) Let p > 2. If x̃ is the principal solution then (22) holds.
∞
∞ 1−q
r (t) dt = ∞, the function γ(t) := t c(s) ds exists and
(iii) Suppose that
γ(t) 0, but γ(t) ≡ 0 eventually. Then x̃(t) is the principal solution if and only
if (22) holds.
189
This theorem shows that the equivalent integral characterization of the principal
solution of (1) is known only in some particular cases. The subject of the present
investigation is whether (22) is really a good characterization. Another intensively
studied problem is the limit characterization of the principal solution of (1), i.e. the
extension of the first relation of (21) to solutions of (1). For some results of this
effort see [8], [15], [19].
4. Related topics
In this section we very briefly present some selected results related to the oscillation
theory of (1).
4.1.
. In this subsection
we suppose that the function c in (1) is positive or negative for large t, in the former
case we suppose in addition that (1) is nonoscillatory. Since the solution space of (1)
is homogeneous, we can restrict our attention to positive solutions. These solutions
can be divided into two main classes according to their behaviour for large t,
Å+
and each class
Å+
Å−
= {x : x (t) > 0},
Å +, Å −
+
=Å+
∞ ∪ Å B,
−
=Å−
B ∪Å0 ,
Å−
= {x : x (t) < 0},
is the union of two subclasses
Å +∞ = {x : x(t) → ∞}, Å +B = {x : x(t) → L < ∞},
Å −B = {x : x(t) → L > 0}, Å −0 = {x : x(t) → 0}.
The investigation of the classification of nonoscillatory solutions of linear equations
(2) along this line was initiated in [35]. Afterward, several papers extending the
results of this paper have appeared, see e.g. in [7], [9] and the references given therein.
An important role is played by the integrals
T
T
J1 = lim
T →∞
J2 = lim
T →∞
r1−q (t)Φ−1
r
1−q
(t)Φ
−1
t
T
c(s) ds dt,
c(s) ds dt
t
and using these intgrals the following results have been established in [7].
Theorem 7. Suppose that c(t) < 0 for large t, then
(a) J1 = −∞, J2 = −∞ =⇒ Å − = Å −
0 = ∅,
−
(b) J1 = −∞, J2 > −∞ =⇒ Å = Å −
B = ∅,
190
(c) J1 > −∞, J2 > −∞ =⇒ Å −
B = ∅,
(d) J1 = −∞ =⇒ Å + = Å +
,
∞
(e) J1 > −∞ =⇒ Å + = Å +
B.
Å −0
(1).
4.2.
problem
(23)
= ∅,
Consider the Dirichlet boundary value
(Φ(x )) + λΦ(x) + g(t, x) = f (t),
x(0) = 0 = x(πp ),
where λ is a real-valued parameter, the number πp is defined in Section 2 and the
“nonhalf-linearity” g and the forcing term f satisfy certain additional assumptions.
The literature dealing with solvability of (23) (not only with the Dirichlet boundary
condition) is very voluminous, see e.g. [10] and the references given therein. The
situation is similar to the linear case in some aspects, but in some cases one meets
completely different phenomena. Here we mention one of them, the Fredholm-type
alternative for solvability of the BVP
(24)
(Φ(x )) + λΦ(x) = f (t),
x(0) = 0 = x(πp ).
Note that λ1 = (p − 1) is the principal eigenvalue of the unforced equation (24).
Using the detailed analysis of the geometry of the functional associated with (24)
Jf (y) =
1
p
πp
{|y |p − λ|y|p } dt −
0
πp
f (t)y dt
0
whose critical points over W01,p (0, πp ) are solutions of (24), one can prove the following statements, see [20] and the references given therein.
Theorem 8. Fredholm’s alternative for p-Laplacian with p = 2.
(i) If λ < 0, the functional Jf has a unique minimum over W01,p (0, πp ), it is coercive
and (24) has a unique solution.
(ii) If 0 < λ < λ1 , the functional Jf is still coercive, but there exists f ∈ C[0, πp ]
such that Jf has at least two critical points (one of them is the global minimum
over W01,p (0, πp ), the other one is of saddle type).
(iii) If λ = λ1 (and similarly for higher eigenvalues λn = (p − 1)n ), the condition
πp
f (t)S(t) dt = 0,
0
where S(t) is the generalized sine function given in Section 2, i.e. the solution of
(24) with f ≡ 0, is only sufficient but not necessary for solvability of (24). More
191
precisely, there exists an open cone C ⊂ C[0, πp ] such that (24) has at least two
solutions for every f ∈ C and
πp
f (t)S(t) dt = 0.
0
4.3.
. In recent years, considerable
attention has been paid to the oscillation theory of various difference equations. In
particular, the oscillation theory of the Sturm-Louville difference equation
∆(rk ∆xk ) + ck xk+1 = 0,
∆xk = xk+1 − xk ,
has been deeply developed, see [1], [31]. The discrete counterpart of (1) is the
difference equation
(25)
∆(rk Φ(∆xk )) + ck Φ(xk+1 ) = 0,
rk = 0.
The basic oscillatory properties of (25) are established in [38] and the main result of
that paper is the discrete version of the Roundabout Theorem (Proposition 1).
Theorem 9. The following statements are equivalent
(i) Equation (25) is disconjugate on the discrete interval [0, N ], i.e. the solution x
given by the initial condition x0 = 0, x1 = 0 has no generalized zero in (0, N +1],
i.e.
rk xk xk+1 > 0, k = 1, . . . , N.
(ii) There exists a solution x of (25) having no generalized zero in [0, N + 1].
+1
(iii) There exists a solution w = {wk }N
k=0 of the Riccati-type equation
∆wk + ck + 1 −
rk
wk = 0,
Φ(Φ−1 (rk ) + Φ−1 (wk ))
w=
rΦ(∆x)
Φ(x)
such that rk + wk > 0, k = 1, . . . , N .
(iv) We have
N
Fd (y; 0, N ) =
{rk |∆yk |p − ck |yk+1 |p } > 0
k=0
+1
for every nontrivial y = {yk }N
k=0 satisfying y0 = 0 = yN +1 .
p. Several physical phenomena can
4.4.
be described by the partial differential equation with the so-called p-Laplacian
(26)
192
div ∇up−2 ∇u + c(x)Φ(u) = 0,
x = (x1 , . . . , xn ) ∈ Ên ,
see e.g. [11]. It can be shown that the p-degree functional
F (u; Ω) =
Ω
{∇up − c(x)|u|p } dx,
the Riccati type equation
(27)
div w + c(x) + (p − 1)wq = 0,
w=
∇u
u
and Picone’s-type identity
F (y; Ω) =
∂Ω
P (u, v) =
w|y|p dS + p
P (∇y, Φ(y)w) dx,
Ω
vq
up
− u, v +
0,
p
q
where w is a solution of (27) defined in the whole domain Ω, can be used to establish
the Roundabout Theorem, the Sturmian theory and to a certain extent also the
oscillation theory similarly as for the ordinary differential equation (1) and for partial
differential equations with the “normal” Laplacian, see [2], [3], [18], [29].
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Author’s address: Ondřej Došlý, Mathematical Institute, Czech Academy of Sciences,
Žižkova 22, CZ-616 62 Brno, e-mail: dosly@math.muni.cz.
195